As the bill to rewrite U.S. immigration laws edges closer to a vote in the Senate, its architects are wrestling with a familiar dilemma: how far should they go to mollify critics and attract new supporters, without wrecking the bipartisan compromise that took months to forge?

In the days since the landmark legislation made its way to the Senate floor, border security has emerged as a flashpoint. GOP Senator Marco Rubio, one of eight negotiators to craft the legislation and its chief envoy to conservatives, says the bill can’t pass without tougher border security. While Democrats aren’t happy about Rubio’s pronouncement, they’re willing to make concessions to lure teetering members and win a striking bipartisan majority that could pave the way for passage in the Republican-controlled House.

But despite ongoing discussions, the two sides have yet to find a sweet spot between Republicans’ desire for tougher security standards and negotiators’ insistence on preserving the core of the bill.

Last week much of the GOP rallied around an amendment filed by John Cornyn of Texas, the No. 2 Senate Republican, which would block undocumented immigrants from applying for green cards until government agencies certify that the border is secure. Cornyn’s amendment has stringent standards for measuring a secure border, including a 90% apprehension rate for illegal crossings, an operational biometric ID system and the capability to fully monitor the 1,900-mile Southern border. Supporters balked, arguing the plan would jeopardize the path to citizenship, an essential component of the bill. Both Democrats and Republicans called the proposal a “poison pill.”

Cornyn tells TIME he hopes his amendment will receive a vote on Wednesday, though he acknowledged it didn’t have the votes to pass yet. “One non-negotiable item,” he says, is tying the legalization process to tougher border enforcement, “which guarantees that everybody is highly incentivized to get the border security piece done. Absent that, I think it’s just another hollow promise.”

Since this is a common refrain on the GOP side, negotiators have been hunting for an alternative that can satisfy Republicans without squandering Democratic support. Late Monday, they began zeroing in on a similar proposal crafted by a pair of GOP Senators, Tennessee’s Bob Corker and John Hoeven of North Dakota. Their amendment, whose language has not been introduced, would lay out similarly tough but perhaps achievable standards for workplace enforcement and border security.

In interviews at the Capitol on Tuesday, senators involved in, or briefed on, the talks declined to go into specifics about the negotiations. “By tomorrow, we will have either signs of progress or a lack of progress,” says Republican Senator John McCain. But his colleague Lindsey Graham of South Carolina told TIME the proposal held promise. “I know the difference between an amendment that’s offered by somebody who will never get to yes versus somebody who could,” says Graham, who like McCain is a member of the so-called Gang of Eight that negotiated the immigration bill. “I think they want to get to yes if they can.”

“I think there is room for border security improvement,” Graham added, thronged by reporters outside a room in the Capitol where his colleagues were helping themselves to a lunch buffet. But it has to meet two conditions, he says: “It has to be affordable, and any new trigger has to be achievable.”

Senators from both parties couldn’t say Tuesday whether the emerging alternative to Cornyn’s proposal would be any more palatable to Democrats. Supporters of immigration reform are leery of stalling tactics, such as senators who dangle their support in order to open sluggish negotiations designed to stymie the bill’s progress. Democrats are hoping to avoid a protracted floor fight that kindles the kind of fervent grassroots opposition to the bill that has yet to materialize. And they fear a lengthy battle could also create a legislative bottleneck — “as anyone who’s been here 10 minutes knows,” says Missouri Democrat Claire McCaskill — crowding out other time-sensitive priorities like student loans in the process.

Not everyone appears interested in appeasing Republicans. Senator Harry Reid, the Democratic leader, says the bipartisan bill could pass the Senate now. While supporters hope to win passage of the measure by July 4, Reid is threatening to file cloture in the coming days to wrap up debate and hold a final vote — with or without wavering Republicans onboard.

But the members who negotiated the measure aren’t keen to ram it through the Senate. Eking out a narrow win on a near-party line vote, they reason, would only stiffen opposition on the other side of the Capitol. Instead, they hope to amass a formidable bipartisan majority that imbues the bill with momentum as it heads to the Republican-controlled House. Speaker John Boehner said Tuesday he doesn’t “see any way” to move a bill without support from a majority of Republicans. But senators are betting that a formidable margin would marshal national pressure and force his hand. “We’re trying to get as many votes as possible,” McCain says.

“Border security is the key to Republican votes. It’s the key to my vote,” says Republican Senator Lamar Alexander of Tennessee. “Senator Corker, Senator Hoeven and others are looking for additional ways to do that that Democrats can accept. Senator Cornyn has come up with a plan that many Democrats cannot accept. So we’re continuing to work on that.”